/*
 * The MIT License
 *
 * Copyright 2013-2014 Florian Barras.
 *
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in
 * all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
 * THE SOFTWARE.
 */
package jeo.math.linearalgebra;

import java.io.Serializable;

/**
 * Eigenvalues and eigenvectors of a real matrix.
 * <p>
 * If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal
 * and the eigenvector matrix V is orthogonal. I.e. A =
 * V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the
 * identity matrix.
 * <p>
 * If A is not symmetric, then the eigenvalue matrix D is block diagonal with
 * the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda +
 * i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent
 * the eigenvectors in the sense that A*V = V*D, i.e. A.times(V) equals
 * V.times(D). The matrix V may be badly conditioned, or even singular, so the
 * validity of the equation A = V*D*inverse(V) depends upon V.cond().
 * <p>
 * @author JAMA, http://math.nist.gov/javanumerics/jama/
 * @version 1.0.3
 */
public class EigenvalueDecomposition
	implements Serializable
{
	////////////////////////////////////////////////////////////////////////////
	// ATTRIBUTE(S)
	////////////////////////////////////////////////////////////////////////////

	/**
	 * Generated serial version ID.
	 */
	private static final long serialVersionUID = -8801368870718892784L;
	/**
	 * Row and column dimension (square matrix).
	 * <p>
	 * @serial matrix dimension
	 */
	private final int n;
	/**
	 * Arrays for internal storage of eigenvalues.
	 * <p>
	 * @serial internal storage of eigenvalues
	 */
	private final double[] d, e;
	/**
	 * Array for internal storage of eigenvectors.
	 * <p>
	 * @serial internal storage of eigenvectors
	 */
	private final double[][] V;
	/**
	 * Array for internal storage of nonsymmetric Hessenberg form.
	 * <p>
	 * @serial internal storage of nonsymmetric Hessenberg form
	 */
	private double[][] H;
	/**
	 * Working storage for nonsymmetric algorithm.
	 * <p>
	 * @serial working storage for nonsymmetric algorithm
	 */
	private double[] ort;
	/**
	 * Symmetry flag.
	 * <p>
	 * @serial internal symmetry flag
	 */
	private boolean issymmetric;
	/**
	 * Complex scalar division.
	 */
	private transient double cdivr, cdivi;


	////////////////////////////////////////////////////////////////////////////
	// CONSTRUCTOR(S)
	////////////////////////////////////////////////////////////////////////////

	/**
	 * Checks for symmetry, then constructs the eigenvalue decomposition.
	 * <p>
	 * Note: structure to access {@code D} and {@code V}.
	 * <p>
	 * @param A a square {@code Matrix}
	 */
	public EigenvalueDecomposition(final Matrix A)
	{
		// Initialize
		final double[][] components = A.getComponents();
		n = A.getColumnDimension();
		issymmetric = true;
		for (int j = 0; j < n & issymmetric; ++j)
		{
			for (int i = 0; i < n & issymmetric; ++i)
			{
				issymmetric = components[i][j] == components[j][i];
			}
		}
		d = new double[n];
		e = new double[n];
		V = new double[n][n];
		// Main loop
		if (issymmetric)
		{
			for (int i = 0; i < n; ++i)
			{
				System.arraycopy(components[i], 0, V[i], 0, n);
			}
			// Tridiagonalize
			tred2();
			// Diagonalize
			tql2();
		}
		else
		{
			H = new double[n][n];
			ort = new double[n];
			for (int j = 0; j < n; ++j)
			{
				for (int i = 0; i < n; ++i)
				{
					H[i][j] = components[i][j];
				}
			}
			// Reduce to Hessenberg form
			orthes();
			// Reduce Hessenberg to real Schur form
			hqr2();
		}
	}


	////////////////////////////////////////////////////////////////////////////
	// GETTER(S)
	////////////////////////////////////////////////////////////////////////////

	/**
	 * Returns the eigenvector matrix {@code V}.
	 * <p>
	 * @return the eigenvector matrix {@code V}
	 */
	public Matrix getV()
	{
		return new Matrix(n, n, V);
	}

	/**
	 * Returns the real parts of the eigenvalues.
	 * <p>
	 * @return real(diag({@code D}))
	 */
	public double[] getRealEigenvalues()
	{
		return d;
	}

	/**
	 * Returns the imaginary parts of the eigenvalues.
	 * <p>
	 * @return imag(diag({@code D}))
	 */
	public double[] getImagEigenvalues()
	{
		return e;
	}

	/**
	 * Returns the block diagonal eigenvalue matrix {@code D}.
	 * <p>
	 * @return the block diagonal eigenvalue matrix {@code D}
	 */
	public Matrix getD()
	{
		final Matrix X = new Matrix(n, n);
		final double[][] D = X.getComponents();
		for (int i = 0; i < n; ++i)
		{
			for (int j = 0; j < n; ++j)
			{
				D[i][j] = 0.0;
			}
			D[i][i] = d[i];
			if (e[i] > 0)
			{
				D[i][i + 1] = e[i];
			}
			else if (e[i] < 0)
			{
				D[i][i - 1] = e[i];
			}
		}
		return X;
	}


	////////////////////////////////////////////////////////////////////////////
	// EIGENVALUE DECOMPOSITION
	////////////////////////////////////////////////////////////////////////////

	/**
	 * Symmetric Householder reduction to tridiagonal form.
	 * <p>
	 * Note: this is derived from the Algol procedures tred2 by Bowdler, Martin,
	 * Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra,
	 * and the corresponding Fortran subroutine in EISPACK.
	 */
	private void tred2()
	{
		System.arraycopy(V[n - 1], 0, d, 0, n);
		// Householder reduction to tridiagonal form
		for (int i = n - 1; i > 0; --i)
		{
			// Scale to avoid under/overflow
			double scale = 0.0;
			double h = 0.0;
			for (int k = 0; k < i; ++k)
			{
				scale += Math.abs(d[k]);
			}
			if (scale == 0.0)
			{
				e[i] = d[i - 1];
				for (int j = 0; j < i; ++j)
				{
					d[j] = V[i - 1][j];
					V[i][j] = 0.0;
					V[j][i] = 0.0;
				}
			}
			else
			{
				// Generate Householder vector
				for (int k = 0; k < i; ++k)
				{
					d[k] /= scale;
					h += d[k] * d[k];
				}
				double f = d[i - 1];
				double g = Math.sqrt(h);
				if (f > 0)
				{
					g = -g;
				}
				e[i] = scale * g;
				h -= f * g;
				d[i - 1] = f - g;
				for (int j = 0; j < i; ++j)
				{
					e[j] = 0.0;
				}
				// Apply similarity transformation to remaining columns
				for (int j = 0; j < i; ++j)
				{
					f = d[j];
					V[j][i] = f;
					g = e[j] + V[j][j] * f;
					for (int k = j + 1; k <= i - 1; ++k)
					{
						g += V[k][j] * d[k];
						e[k] += V[k][j] * f;
					}
					e[j] = g;
				}
				f = 0.0;
				for (int j = 0; j < i; ++j)
				{
					e[j] /= h;
					f += e[j] * d[j];
				}
				final double hh = f / (h + h);
				for (int j = 0; j < i; ++j)
				{
					e[j] -= hh * d[j];
				}
				for (int j = 0; j < i; ++j)
				{
					f = d[j];
					g = e[j];
					for (int k = j; k <= i - 1; ++k)
					{
						V[k][j] -= f * e[k] + g * d[k];
					}
					d[j] = V[i - 1][j];
					V[i][j] = 0.0;
				}
			}
			d[i] = h;
		}
		// Accumulate transformations
		for (int i = 0; i < n - 1; ++i)
		{
			V[n - 1][i] = V[i][i];
			V[i][i] = 1.0;
			final double h = d[i + 1];
			if (h != 0.0)
			{
				for (int k = 0; k <= i; ++k)
				{
					d[k] = V[k][i + 1] / h;
				}
				for (int j = 0; j <= i; ++j)
				{
					double g = 0.0;
					for (int k = 0; k <= i; ++k)
					{
						g += V[k][i + 1] * V[k][j];
					}
					for (int k = 0; k <= i; ++k)
					{
						V[k][j] -= g * d[k];
					}
				}
			}
			for (int k = 0; k <= i; ++k)
			{
				V[k][i + 1] = 0.0;
			}
		}
		for (int j = 0; j < n; ++j)
		{
			d[j] = V[n - 1][j];
			V[n - 1][j] = 0.0;
		}
		V[n - 1][n - 1] = 1.0;
		e[0] = 0.0;
	}

	/**
	 * Symmetric tridiagonal QL algorithm.
	 * <p>
	 * Note: this is derived from the Algol procedures tql2, by Bowdler, Martin,
	 * Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra,
	 * and the corresponding Fortran subroutine in EISPACK.
	 */
	private void tql2()
	{
		for (int i = 1; i < n; ++i)
		{
			e[i - 1] = e[i];
		}
		e[n - 1] = 0.0;
		double f = 0.0;
		double tst1 = 0.0;
		final double eps = Math.pow(2.0, -52.0);
		for (int l = 0; l < n; ++l)
		{
			// Find small subdiagonal element
			tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l]));
			int m = l;
			while (m < n)
			{
				if (Math.abs(e[m]) <= eps * tst1)
				{
					break;
				}
				++m;
			}
			// If m == l, d[l] is an eigenvalue,
			// otherwise, iterate
			if (m > l)
			{
				// int iter = 0;
				do
				{
					// iter += 1;
					// Note: could check iteration count here
					// Compute implicit shift
					double g = d[l];
					double p = (d[l + 1] - g) / (2.0 * e[l]);
					double r = Norms.getEuclideanNorm(p, 1.0);
					if (p < 0)
					{
						r = -r;
					}
					d[l] = e[l] / (p + r);
					d[l + 1] = e[l] * (p + r);
					final double dl1 = d[l + 1];
					double h = g - d[l];
					for (int i = l + 2; i < n; ++i)
					{
						d[i] -= h;
					}
					f += h;
					// Implicit QL transformation
					p = d[m];
					double c = 1.0;
					double c2 = c;
					double c3 = c;
					final double el1 = e[l + 1];
					double s = 0.0;
					double s2 = 0.0;
					for (int i = m - 1; i >= l; --i)
					{
						c3 = c2;
						c2 = c;
						s2 = s;
						g = c * e[i];
						h = c * p;
						r = Norms.getEuclideanNorm(p, e[i]);
						e[i + 1] = s * r;
						s = e[i] / r;
						c = p / r;
						p = c * d[i] - s * g;
						d[i + 1] = h + s * (c * g + s * d[i]);
						// Accumulate transformation
						for (int k = 0; k < n; ++k)
						{
							h = V[k][i + 1];
							V[k][i + 1] = s * V[k][i] + c * h;
							V[k][i] = c * V[k][i] - s * h;
						}
					}
					p = -s * s2 * c3 * el1 * e[l] / dl1;
					e[l] = s * p;
					d[l] = c * p;
				}
				// Check for convergence
				while (Math.abs(e[l]) > eps * tst1);
			}
			d[l] += f;
			e[l] = 0.0;
		}
		// Sort eigenvalues and corresponding vectors
		for (int i = 0; i < n - 1; ++i)
		{
			int k = i;
			double p = d[i];
			for (int j = i + 1; j < n; ++j)
			{
				if (d[j] < p)
				{
					k = j;
					p = d[j];
				}
			}
			if (k != i)
			{
				d[k] = d[i];
				d[i] = p;
				for (int j = 0; j < n; ++j)
				{
					p = V[j][i];
					V[j][i] = V[j][k];
					V[j][k] = p;
				}
			}
		}
	}

	/**
	 * Nonsymmetric reduction to Hessenberg form.
	 * <p>
	 * Note: this is derived from the Algol procedures orthes and ortran, by
	 * Martin and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra,
	 * and the corresponding Fortran subroutines in EISPACK.
	 */
	private void orthes()
	{
		final int low = 0;
		final int high = n - 1;
		for (int m = low + 1; m <= high - 1; ++m)
		{
			// Scale column
			double scale = 0.0;
			for (int i = m; i <= high; ++i)
			{
				scale += Math.abs(H[i][m - 1]);
			}
			if (scale != 0.0)
			{
				// Compute Householder transformation
				double h = 0.0;
				for (int i = high; i >= m; --i)
				{
					ort[i] = H[i][m - 1] / scale;
					h += ort[i] * ort[i];
				}
				double g = Math.sqrt(h);
				if (ort[m] > 0)
				{
					g = -g;
				}
				h -= ort[m] * g;
				ort[m] -= g;
				// Apply Householder similarity transformation
				// H = (I-u*u'/h)*H*(I-u*u')/h)
				for (int j = m; j < n; ++j)
				{
					double f = 0.0;
					for (int i = high; i >= m; --i)
					{
						f += ort[i] * H[i][j];
					}
					f /= h;
					for (int i = m; i <= high; ++i)
					{
						H[i][j] -= f * ort[i];
					}
				}
				for (int i = 0; i <= high; ++i)
				{
					double f = 0.0;
					for (int j = high; j >= m; --j)
					{
						f += ort[j] * H[i][j];
					}
					f /= h;
					for (int j = m; j <= high; ++j)
					{
						H[i][j] -= f * ort[j];
					}
				}
				ort[m] = scale * ort[m];
				H[m][m - 1] = scale * g;
			}
		}
		// Accumulate transformations (Algol's ortran)
		for (int i = 0; i < n; ++i)
		{
			for (int j = 0; j < n; ++j)
			{
				V[i][j] = i == j ? 1.0 : 0.0;
			}
		}
		for (int m = high - 1; m >= low + 1; --m)
		{
			if (H[m][m - 1] != 0.0)
			{
				for (int i = m + 1; i <= high; ++i)
				{
					ort[i] = H[i][m - 1];
				}
				for (int j = m; j <= high; ++j)
				{
					double g = 0.0;
					for (int i = m; i <= high; ++i)
					{
						g += ort[i] * V[i][j];
					}
					// Double division avoids possible underflow
					g = g / ort[m] / H[m][m - 1];
					for (int i = m; i <= high; ++i)
					{
						V[i][j] += g * ort[i];
					}
				}
			}
		}
	}

	/**
	 * Complex scalar division.
	 */
	private void cdiv(final double xr, final double xi, final double yr, final double yi)
	{
		final double r, d;
		if (Math.abs(yr) > Math.abs(yi))
		{
			r = yi / yr;
			d = yr + r * yi;
			cdivr = (xr + r * xi) / d;
			cdivi = (xi - r * xr) / d;
		}
		else
		{
			r = yr / yi;
			d = yi + r * yr;
			cdivr = (r * xr + xi) / d;
			cdivi = (r * xi - xr) / d;
		}
	}

	/**
	 * Nonsymmetric reduction from Hessenberg to real Schur form.
	 * <p>
	 * Note: this is derived from the Algol procedure hqr2, by Martin and
	 * Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the
	 * corresponding Fortran subroutine in EISPACK.
	 */
	private void hqr2()
	{
		// Initialize
		final int nn = n;
		int n = nn - 1;
		final int low = 0;
		final int high = nn - 1;
		// Floating-point relative accuracy
		final double eps = Math.pow(2.0, -52.0);
		double exshift = 0.0;
		double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;
		// Store roots isolated by balanc and compute matrix norm
		double norm = 0.0;
		for (int i = 0; i < nn; ++i)
		{
			if (i < low | i > high)
			{
				d[i] = H[i][i];
				e[i] = 0.0;
			}
			for (int j = Math.max(i - 1, 0); j < nn; ++j)
			{
				norm += Math.abs(H[i][j]);
			}
		}
		// Outer loop over eigenvalue index
		int iter = 0;
		while (n >= low)
		{
			// Look for single small sub-diagonal element
			int l = n;
			while (l > low)
			{
				s = Math.abs(H[l - 1][l - 1]) + Math.abs(H[l][l]);
				if (s == 0.0)
				{
					s = norm;
				}
				if (Math.abs(H[l][l - 1]) < eps * s)
				{
					break;
				}
				--l;
			}
			// Check for convergence
			// - 1 root found
			if (l == n)
			{
				H[n][n] += exshift;
				d[n] = H[n][n];
				e[n] = 0.0;
				--n;
				iter = 0;
			}
			// - 2 roots found
			else if (l == n - 1)
			{
				w = H[n][n - 1] * H[n - 1][n];
				p = (H[n - 1][n - 1] - H[n][n]) / 2.0;
				q = p * p + w;
				z = Math.sqrt(Math.abs(q));
				H[n][n] += exshift;
				H[n - 1][n - 1] += exshift;
				x = H[n][n];
				// Real pair
				if (q >= 0)
				{
					if (p >= 0)
					{
						z = p + z;
					}
					else
					{
						z = p - z;
					}
					d[n - 1] = x + z;
					d[n] = d[n - 1];
					if (z != 0.0)
					{
						d[n] = x - w / z;
					}
					e[n - 1] = 0.0;
					e[n] = 0.0;
					x = H[n][n - 1];
					s = Math.abs(x) + Math.abs(z);
					p = x / s;
					q = z / s;
					r = Math.sqrt(p * p + q * q);
					p /= r;
					q /= r;
					// Row modification
					for (int j = n - 1; j < nn; ++j)
					{
						z = H[n - 1][j];
						H[n - 1][j] = q * z + p * H[n][j];
						H[n][j] = q * H[n][j] - p * z;
					}
					// Column modification
					for (int i = 0; i <= n; ++i)
					{
						z = H[i][n - 1];
						H[i][n - 1] = q * z + p * H[i][n];
						H[i][n] = q * H[i][n] - p * z;
					}
					// Accumulate transformations
					for (int i = low; i <= high; ++i)
					{
						z = V[i][n - 1];
						V[i][n - 1] = q * z + p * V[i][n];
						V[i][n] = q * V[i][n] - p * z;
					}
				}
				// Complex pair
				else
				{
					d[n - 1] = x + p;
					d[n] = x + p;
					e[n - 1] = z;
					e[n] = -z;
				}
				n -= 2;
				iter = 0;
			}
			// No convergence yet
			else
			{
				// Form shift
				x = H[n][n];
				y = 0.0;
				w = 0.0;
				if (l < n)
				{
					y = H[n - 1][n - 1];
					w = H[n][n - 1] * H[n - 1][n];
				}
				// Wilkinson's original ad hoc shift
				if (iter == 10)
				{
					exshift += x;
					for (int i = low; i <= n; ++i)
					{
						H[i][i] -= x;
					}
					s = Math.abs(H[n][n - 1]) + Math.abs(H[n - 1][n - 2]);
					x = y = 0.75 * s;
					w = -0.4375 * s * s;
				}
				// MATLAB's new ad hoc shift
				if (iter == 30)
				{
					s = (y - x) / 2.0;
					s = s * s + w;
					if (s > 0)
					{
						s = Math.sqrt(s);
						if (y < x)
						{
							s = -s;
						}
						s = x - w / ((y - x) / 2.0 + s);
						for (int i = low; i <= n; ++i)
						{
							H[i][i] -= s;
						}
						exshift += s;
						x = y = w = 0.964;
					}
				}
				iter += 1;
				// Note: could check iteration count here
				// Look for two consecutive small sub-diagonal elements
				int m = n - 2;
				while (m >= l)
				{
					z = H[m][m];
					r = x - z;
					s = y - z;
					p = (r * s - w) / H[m + 1][m] + H[m][m + 1];
					q = H[m + 1][m + 1] - z - r - s;
					r = H[m + 2][m + 1];
					s = Math.abs(p) + Math.abs(q) + Math.abs(r);
					p /= s;
					q /= s;
					r /= s;
					if (m == l)
					{
						break;
					}
					if (Math.abs(H[m][m - 1]) * (Math.abs(q) + Math.abs(r)) < eps * (Math.abs(p) * (Math.abs(H[m - 1][m - 1]) + Math.abs(z) + Math.abs(H[m + 1][m + 1]))))
					{
						break;
					}
					--m;
				}
				for (int i = m + 2; i <= n; ++i)
				{
					H[i][i - 2] = 0.0;
					if (i > m + 2)
					{
						H[i][i - 3] = 0.0;
					}
				}
				// Double QR step involving rows l:n and columns m:n
				for (int k = m; k <= n - 1; ++k)
				{
					final boolean notlast = k != n - 1;
					if (k != m)
					{
						p = H[k][k - 1];
						q = H[k + 1][k - 1];
						r = notlast ? H[k + 2][k - 1] : 0.0;
						x = Math.abs(p) + Math.abs(q) + Math.abs(r);
						if (x == 0.0)
						{
							continue;
						}
						p /= x;
						q /= x;
						r /= x;
					}
					s = Math.sqrt(p * p + q * q + r * r);
					if (p < 0)
					{
						s = -s;
					}
					if (s != 0)
					{
						if (k != m)
						{
							H[k][k - 1] = -s * x;
						}
						else if (l != m)
						{
							H[k][k - 1] = -H[k][k - 1];
						}
						p += s;
						x = p / s;
						y = q / s;
						z = r / s;
						q /= p;
						r /= p;
						// Row modification
						for (int j = k; j < nn; ++j)
						{
							p = H[k][j] + q * H[k + 1][j];
							if (notlast)
							{
								p += r * H[k + 2][j];
								H[k + 2][j] -= p * z;
							}
							H[k][j] -= p * x;
							H[k + 1][j] -= p * y;
						}
						// Column modification
						for (int i = 0; i <= Math.min(n, k + 3); ++i)
						{
							p = x * H[i][k] + y * H[i][k + 1];
							if (notlast)
							{
								p += z * H[i][k + 2];
								H[i][k + 2] -= p * r;
							}
							H[i][k] -= p;
							H[i][k + 1] -= p * q;
						}
						// Accumulate transformations
						for (int i = low; i <= high; ++i)
						{
							p = x * V[i][k] + y * V[i][k + 1];
							if (notlast)
							{
								p += z * V[i][k + 2];
								V[i][k + 2] -= p * r;
							}
							V[i][k] -= p;
							V[i][k + 1] -= p * q;
						}
					} // (s != 0)
				} // k loop
			} // check convergence
		} // while (n >= low)
		// Backsubstitute to find vectors of upper triangular form
		if (norm == 0.0)
		{
			return;
		}
		for (n = nn - 1; n >= 0; --n)
		{
			p = d[n];
			q = e[n];
			// Real vector
			if (q == 0)
			{
				int l = n;
				H[n][n] = 1.0;
				for (int i = n - 1; i >= 0; --i)
				{
					w = H[i][i] - p;
					r = 0.0;
					for (int j = l; j <= n; ++j)
					{
						r += H[i][j] * H[j][n];
					}
					if (e[i] < 0.0)
					{
						z = w;
						s = r;
					}
					else
					{
						l = i;
						if (e[i] == 0.0)
						{
							if (w != 0.0)
							{
								H[i][n] = -r / w;
							}
							else
							{
								H[i][n] = -r / (eps * norm);
							}
						}
						// Solve real equations
						else
						{
							x = H[i][i + 1];
							y = H[i + 1][i];
							q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
							t = (x * s - z * r) / q;
							H[i][n] = t;
							if (Math.abs(x) > Math.abs(z))
							{
								H[i + 1][n] = (-r - w * t) / x;
							}
							else
							{
								H[i + 1][n] = (-s - y * t) / z;
							}
						}
						// Overflow control
						t = Math.abs(H[i][n]);
						if (eps * t * t > 1)
						{
							for (int j = i; j <= n; ++j)
							{
								H[j][n] /= t;
							}
						}
					}
				}
			}
			// Complex vector
			else if (q < 0)
			{
				int l = n - 1;
				// Last vector component imaginary so matrix is triangular
				if (Math.abs(H[n][n - 1]) > Math.abs(H[n - 1][n]))
				{
					H[n - 1][n - 1] = q / H[n][n - 1];
					H[n - 1][n] = -(H[n][n] - p) / H[n][n - 1];
				}
				else
				{
					cdiv(0.0, -H[n - 1][n], H[n - 1][n - 1] - p, q);
					H[n - 1][n - 1] = cdivr;
					H[n - 1][n] = cdivi;
				}
				H[n][n - 1] = 0.0;
				H[n][n] = 1.0;
				for (int i = n - 2; i >= 0; --i)
				{
					double ra, sa, vr, vi;
					ra = 0.0;
					sa = 0.0;
					for (int j = l; j <= n; ++j)
					{
						ra += H[i][j] * H[j][n - 1];
						sa += H[i][j] * H[j][n];
					}
					w = H[i][i] - p;
					if (e[i] < 0.0)
					{
						z = w;
						r = ra;
						s = sa;
					}
					else
					{
						l = i;
						if (e[i] == 0)
						{
							cdiv(-ra, -sa, w, q);
							H[i][n - 1] = cdivr;
							H[i][n] = cdivi;
						}
						else
						{
							// Solve complex equations
							x = H[i][i + 1];
							y = H[i + 1][i];
							vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
							vi = (d[i] - p) * 2.0 * q;
							if (vr == 0.0 & vi == 0.0)
							{
								vr = eps * norm * (Math.abs(w) + Math.abs(q) + Math.abs(x) + Math.abs(y) + Math.abs(z));
							}
							cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);
							H[i][n - 1] = cdivr;
							H[i][n] = cdivi;
							if (Math.abs(x) > Math.abs(z) + Math.abs(q))
							{
								H[i + 1][n - 1] = (-ra - w * H[i][n - 1] + q * H[i][n]) / x;
								H[i + 1][n] = (-sa - w * H[i][n] - q * H[i][n - 1]) / x;
							}
							else
							{
								cdiv(-r - y * H[i][n - 1], -s - y * H[i][n], z, q);
								H[i + 1][n - 1] = cdivr;
								H[i + 1][n] = cdivi;
							}
						}
						// Overflow control
						t = Math.max(Math.abs(H[i][n - 1]), Math.abs(H[i][n]));
						if (eps * t * t > 1)
						{
							for (int j = i; j <= n; ++j)
							{
								H[j][n - 1] /= t;
								H[j][n] /= t;
							}
						}
					}
				}
			}
		}
		// Vectors of isolated roots
		for (int i = 0; i < nn; ++i)
		{
			if (i < low | i > high)
			{
				System.arraycopy(H[i], i, V[i], i, nn - i);
			}
		}
		// Back transformation to get eigenvectors of original matrix
		for (int j = nn - 1; j >= low; --j)
		{
			for (int i = low; i <= high; ++i)
			{
				z = 0.0;
				for (int k = low; k <= Math.min(j, high); ++k)
				{
					z += V[i][k] * H[k][j];
				}
				V[i][j] = z;
			}
		}
	}
}
